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Properties

Label 18.2.111...000.1
Degree $18$
Signature $[2, 8]$
Discriminant $1.116\times 10^{30}$
Root discriminant \(46.70\)
Ramified primes $2,3,5,7$
Class number $6$ (GRH)
Class group [6] (GRH)
Galois group $C_6:S_3$ (as 18T12)

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Show commands: Magma / Oscar / Pari/GP / SageMath

Normalized defining polynomial

Copy content comment:Define the number field
 
Copy content sage:x = polygen(QQ); K.<a> = NumberField(x^18 - 6*x^17 + 14*x^16 - 12*x^15 + 144*x^13 - 335*x^12 + 900*x^11 - 1050*x^10 + 1470*x^9 - 760*x^8 - 36*x^7 + 127*x^6 - 1038*x^5 + 132*x^4 + 990*x^3 - 504*x^2 + 108*x - 27)
 
Copy content gp:K = bnfinit(y^18 - 6*y^17 + 14*y^16 - 12*y^15 + 144*y^13 - 335*y^12 + 900*y^11 - 1050*y^10 + 1470*y^9 - 760*y^8 - 36*y^7 + 127*y^6 - 1038*y^5 + 132*y^4 + 990*y^3 - 504*y^2 + 108*y - 27, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^18 - 6*x^17 + 14*x^16 - 12*x^15 + 144*x^13 - 335*x^12 + 900*x^11 - 1050*x^10 + 1470*x^9 - 760*x^8 - 36*x^7 + 127*x^6 - 1038*x^5 + 132*x^4 + 990*x^3 - 504*x^2 + 108*x - 27);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^18 - 6*x^17 + 14*x^16 - 12*x^15 + 144*x^13 - 335*x^12 + 900*x^11 - 1050*x^10 + 1470*x^9 - 760*x^8 - 36*x^7 + 127*x^6 - 1038*x^5 + 132*x^4 + 990*x^3 - 504*x^2 + 108*x - 27)
 

\( x^{18} - 6 x^{17} + 14 x^{16} - 12 x^{15} + 144 x^{13} - 335 x^{12} + 900 x^{11} - 1050 x^{10} + \cdots - 27 \) Copy content Toggle raw display

Copy content comment:Defining polynomial
 
Copy content sage:K.defining_polynomial()
 
Copy content gp:K.pol
 
Copy content magma:DefiningPolynomial(K);
 
Copy content oscar:defining_polynomial(K)
 

Invariants

Degree:  $18$
Copy content comment:Degree over Q
 
Copy content sage:K.degree()
 
Copy content gp:poldegree(K.pol)
 
Copy content magma:Degree(K);
 
Copy content oscar:degree(K)
 
Signature:  $[2, 8]$
Copy content comment:Signature
 
Copy content sage:K.signature()
 
Copy content gp:K.sign
 
Copy content magma:Signature(K);
 
Copy content oscar:signature(K)
 
Discriminant:   \(1115906277282951168000000000000\) \(\medspace = 2^{24}\cdot 3^{9}\cdot 5^{12}\cdot 7^{12}\) Copy content Toggle raw display
Copy content comment:Discriminant
 
Copy content sage:K.disc()
 
Copy content gp:K.disc
 
Copy content magma:OK := Integers(K); Discriminant(OK);
 
Copy content oscar:OK = ring_of_integers(K); discriminant(OK)
 
Root discriminant:  \(46.70\)
Copy content sage:(K.disc().abs())^(1./K.degree())
 
Copy content gp:abs(K.disc)^(1/poldegree(K.pol))
 
Copy content magma:Abs(Discriminant(OK))^(1/Degree(K));
 
Copy content oscar:(1.0 * dK)^(1/degree(K))
 
Galois root discriminant:  $2^{4/3}3^{1/2}5^{2/3}7^{2/3}\approx 46.69954520953564$
Ramified primes:   \(2\), \(3\), \(5\), \(7\) Copy content Toggle raw display
Copy content comment:Ramified primes
 
Copy content sage:K.disc().support()
 
Copy content gp:factor(abs(K.disc))[,1]~
 
Copy content magma:PrimeDivisors(Discriminant(OK));
 
Copy content oscar:prime_divisors(discriminant((OK)))
 
Discriminant root field:  \(\Q(\sqrt{3}) \)
$\Aut(K/\Q)$:   $C_2$
Copy content comment:Autmorphisms
 
Copy content sage:K.automorphisms()
 
Copy content magma:Automorphisms(K);
 
Copy content oscar:automorphisms(K)
 
This field is not Galois over $\Q$.
This is not a CM field.
This field has no CM subfields.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{123}a^{14}+\frac{16}{41}a^{13}-\frac{13}{123}a^{12}-\frac{19}{41}a^{11}-\frac{9}{41}a^{10}-\frac{8}{41}a^{9}-\frac{11}{123}a^{8}-\frac{13}{41}a^{7}-\frac{2}{41}a^{6}+\frac{50}{123}a^{4}-\frac{6}{41}a^{3}-\frac{14}{123}a^{2}+\frac{8}{41}a+\frac{4}{41}$, $\frac{1}{1599}a^{15}+\frac{4}{1599}a^{14}-\frac{772}{1599}a^{13}+\frac{23}{1599}a^{12}-\frac{34}{533}a^{11}+\frac{142}{533}a^{10}-\frac{185}{1599}a^{9}-\frac{170}{1599}a^{8}+\frac{6}{41}a^{7}-\frac{158}{533}a^{6}+\frac{419}{1599}a^{5}-\frac{250}{1599}a^{4}-\frac{329}{1599}a^{3}+\frac{517}{1599}a^{2}-\frac{61}{533}a-\frac{12}{533}$, $\frac{1}{897039}a^{16}+\frac{1}{17589}a^{15}+\frac{521}{897039}a^{14}-\frac{47174}{299013}a^{13}+\frac{7444}{27183}a^{12}+\frac{3046}{7667}a^{11}-\frac{229061}{897039}a^{10}+\frac{136912}{299013}a^{9}+\frac{17348}{299013}a^{8}+\frac{116530}{299013}a^{7}-\frac{393061}{897039}a^{6}-\frac{80398}{299013}a^{5}-\frac{163100}{897039}a^{4}+\frac{14429}{99671}a^{3}+\frac{12509}{99671}a^{2}-\frac{18800}{99671}a-\frac{37979}{99671}$, $\frac{1}{62\cdots 97}a^{17}+\frac{1020675559}{62\cdots 97}a^{16}+\frac{1265987400779}{62\cdots 97}a^{15}+\frac{21572351785199}{62\cdots 97}a^{14}-\frac{400656381002309}{20\cdots 99}a^{13}+\frac{77429511719360}{692168241955333}a^{12}+\frac{17\cdots 22}{62\cdots 97}a^{11}-\frac{20\cdots 67}{62\cdots 97}a^{10}-\frac{177402729942247}{20\cdots 99}a^{9}-\frac{814603322678044}{20\cdots 99}a^{8}-\frac{21\cdots 76}{62\cdots 97}a^{7}-\frac{22\cdots 64}{62\cdots 97}a^{6}+\frac{21\cdots 95}{62\cdots 97}a^{5}+\frac{21\cdots 79}{62\cdots 97}a^{4}+\frac{96586800843881}{20\cdots 99}a^{3}-\frac{29123844934163}{188773156896909}a^{2}-\frac{329276049792087}{692168241955333}a+\frac{142874608006781}{692168241955333}$ Copy content Toggle raw display

Copy content comment:Integral basis
 
Copy content sage:K.integral_basis()
 
Copy content gp:K.zk
 
Copy content magma:IntegralBasis(K);
 
Copy content oscar:basis(OK)
 

Monogenic:  No
Index:  Not computed
Inessential primes:  $3$

Class group and class number

Ideal class group:  $C_{6}$, which has order $6$ (assuming GRH)
Copy content comment:Class group
 
Copy content sage:K.class_group().invariants()
 
Copy content gp:K.clgp
 
Copy content magma:ClassGroup(K);
 
Copy content oscar:class_group(K)
 
Narrow class group:  $C_{2}\times C_{6}$, which has order $12$ (assuming GRH)
Copy content comment:Narrow class group
 
Copy content sage:K.narrow_class_group().invariants()
 
Copy content gp:bnfnarrow(K)
 
Copy content magma:NarrowClassGroup(K);
 

Unit group

Copy content comment:Unit group
 
Copy content sage:UK = K.unit_group()
 
Copy content magma:UK, fUK := UnitGroup(K);
 
Copy content oscar:UK, fUK = unit_group(OK)
 
Rank:  $9$
Copy content comment:Unit rank
 
Copy content sage:UK.rank()
 
Copy content gp:K.fu
 
Copy content magma:UnitRank(K);
 
Copy content oscar:rank(UK)
 
Torsion generator:   \( -1 \)  (order $2$) Copy content Toggle raw display
Copy content comment:Generator for roots of unity
 
Copy content sage:UK.torsion_generator()
 
Copy content gp:K.tu[2]
 
Copy content magma:K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
Copy content oscar:torsion_units_generator(OK)
 
Fundamental units:   $\frac{415536079706945}{692168241955333}a^{17}-\frac{63\cdots 16}{20\cdots 99}a^{16}+\frac{11\cdots 85}{20\cdots 99}a^{15}-\frac{24186871405457}{11104303346877}a^{14}-\frac{40\cdots 78}{20\cdots 99}a^{13}+\frac{58\cdots 05}{692168241955333}a^{12}-\frac{87\cdots 40}{692168241955333}a^{11}+\frac{88\cdots 34}{20\cdots 99}a^{10}-\frac{52\cdots 44}{20\cdots 99}a^{9}+\frac{45\cdots 93}{692168241955333}a^{8}+\frac{62\cdots 42}{53243710919641}a^{7}+\frac{88\cdots 91}{122147336815647}a^{6}+\frac{28\cdots 97}{20\cdots 99}a^{5}-\frac{10\cdots 95}{20\cdots 99}a^{4}-\frac{58\cdots 13}{159731132758923}a^{3}+\frac{19\cdots 44}{692168241955333}a^{2}-\frac{21\cdots 19}{40715778938549}a+\frac{12\cdots 01}{692168241955333}$, $\frac{17068241995945}{692168241955333}a^{17}-\frac{228030052237067}{20\cdots 99}a^{16}+\frac{314375025697589}{20\cdots 99}a^{15}+\frac{9323113900829}{159731132758923}a^{14}-\frac{257696307828170}{20\cdots 99}a^{13}+\frac{23\cdots 53}{692168241955333}a^{12}-\frac{19\cdots 08}{692168241955333}a^{11}+\frac{29\cdots 44}{20\cdots 99}a^{10}+\frac{20\cdots 40}{20\cdots 99}a^{9}+\frac{14\cdots 85}{692168241955333}a^{8}+\frac{15\cdots 18}{692168241955333}a^{7}+\frac{17\cdots 71}{20\cdots 99}a^{6}+\frac{16\cdots 53}{20\cdots 99}a^{5}-\frac{34\cdots 16}{20\cdots 99}a^{4}-\frac{53\cdots 67}{188773156896909}a^{3}+\frac{70617779533116}{692168241955333}a^{2}+\frac{32\cdots 03}{692168241955333}a+\frac{4482868923035}{53243710919641}$, $\frac{891903124789982}{62\cdots 97}a^{17}-\frac{270817945303102}{366442010446941}a^{16}+\frac{86\cdots 11}{62\cdots 97}a^{15}-\frac{34\cdots 68}{62\cdots 97}a^{14}-\frac{983580890417980}{20\cdots 99}a^{13}+\frac{14\cdots 66}{692168241955333}a^{12}-\frac{17\cdots 58}{566319470690727}a^{11}+\frac{64\cdots 64}{62\cdots 97}a^{10}-\frac{13\cdots 30}{20\cdots 99}a^{9}+\frac{10\cdots 56}{692168241955333}a^{8}+\frac{13\cdots 76}{566319470690727}a^{7}+\frac{90\cdots 30}{62\cdots 97}a^{6}+\frac{18\cdots 98}{62\cdots 97}a^{5}-\frac{68\cdots 62}{566319470690727}a^{4}-\frac{58\cdots 27}{692168241955333}a^{3}+\frac{14\cdots 08}{20\cdots 99}a^{2}-\frac{88\cdots 48}{692168241955333}a+\frac{193695003411521}{40715778938549}$, $\frac{144488927611882}{151939370185317}a^{17}-\frac{33\cdots 13}{692168241955333}a^{16}+\frac{57\cdots 32}{62\cdots 97}a^{15}-\frac{75\cdots 11}{20\cdots 99}a^{14}-\frac{64\cdots 51}{20\cdots 99}a^{13}+\frac{27\cdots 53}{20\cdots 99}a^{12}-\frac{12\cdots 73}{62\cdots 97}a^{11}+\frac{14\cdots 00}{20\cdots 99}a^{10}-\frac{29\cdots 57}{692168241955333}a^{9}+\frac{72\cdots 22}{692168241955333}a^{8}+\frac{99\cdots 12}{62\cdots 97}a^{7}+\frac{20\cdots 53}{20\cdots 99}a^{6}+\frac{12\cdots 11}{62\cdots 97}a^{5}-\frac{16\cdots 13}{20\cdots 99}a^{4}-\frac{11\cdots 78}{20\cdots 99}a^{3}+\frac{96\cdots 79}{20\cdots 99}a^{2}-\frac{59\cdots 15}{692168241955333}a+\frac{21\cdots 30}{692168241955333}$, $\frac{817524365}{53084457547}a^{17}-\frac{13147860205}{159253372641}a^{16}+\frac{26316097006}{159253372641}a^{15}-\frac{15111959120}{159253372641}a^{14}-\frac{3230122129}{159253372641}a^{13}+\frac{114854913143}{53084457547}a^{12}-\frac{200353651318}{53084457547}a^{11}+\frac{1880273764388}{159253372641}a^{10}-\frac{1530431741447}{159253372641}a^{9}+\frac{1005827907704}{53084457547}a^{8}-\frac{3779876184}{1294742867}a^{7}+\frac{258116926207}{159253372641}a^{6}+\frac{28946501729}{159253372641}a^{5}-\frac{2582275527181}{159253372641}a^{4}-\frac{1222759581359}{159253372641}a^{3}+\frac{414852899238}{53084457547}a^{2}-\frac{71915466701}{53084457547}a+\frac{14034302383}{53084457547}$, $\frac{39\cdots 04}{62\cdots 97}a^{17}-\frac{21\cdots 53}{692168241955333}a^{16}+\frac{34\cdots 92}{62\cdots 97}a^{15}-\frac{32\cdots 32}{20\cdots 99}a^{14}-\frac{43\cdots 05}{20\cdots 99}a^{13}+\frac{18\cdots 94}{20\cdots 99}a^{12}-\frac{44\cdots 41}{366442010446941}a^{11}+\frac{91\cdots 77}{20\cdots 99}a^{10}-\frac{38\cdots 09}{188773156896909}a^{9}+\frac{47\cdots 39}{692168241955333}a^{8}+\frac{14\cdots 73}{62\cdots 97}a^{7}+\frac{30\cdots 14}{20\cdots 99}a^{6}+\frac{11\cdots 55}{62\cdots 97}a^{5}-\frac{10\cdots 61}{20\cdots 99}a^{4}-\frac{31\cdots 31}{692168241955333}a^{3}+\frac{32\cdots 96}{159731132758923}a^{2}-\frac{27\cdots 27}{53243710919641}a+\frac{86\cdots 94}{692168241955333}$, $\frac{17\cdots 30}{20\cdots 99}a^{17}-\frac{20\cdots 52}{479193398276769}a^{16}+\frac{56\cdots 36}{692168241955333}a^{15}-\frac{20\cdots 24}{62\cdots 97}a^{14}-\frac{33\cdots 75}{122147336815647}a^{13}+\frac{63\cdots 64}{53243710919641}a^{12}-\frac{37\cdots 57}{20\cdots 99}a^{11}+\frac{37\cdots 55}{62\cdots 97}a^{10}-\frac{25\cdots 79}{692168241955333}a^{9}+\frac{19\cdots 73}{20\cdots 99}a^{8}+\frac{95\cdots 95}{692168241955333}a^{7}+\frac{53\cdots 10}{62\cdots 97}a^{6}+\frac{37\cdots 13}{20\cdots 99}a^{5}-\frac{45\cdots 20}{62\cdots 97}a^{4}-\frac{10\cdots 67}{20\cdots 99}a^{3}+\frac{85\cdots 94}{20\cdots 99}a^{2}-\frac{52\cdots 01}{692168241955333}a+\frac{18\cdots 32}{692168241955333}$, $\frac{165545487382078}{62\cdots 97}a^{17}-\frac{26413126242943}{188773156896909}a^{16}+\frac{16\cdots 80}{62\cdots 97}a^{15}-\frac{62036611080949}{692168241955333}a^{14}-\frac{337780030619935}{20\cdots 99}a^{13}+\frac{79\cdots 92}{20\cdots 99}a^{12}-\frac{38\cdots 86}{62\cdots 97}a^{11}+\frac{12\cdots 34}{692168241955333}a^{10}-\frac{25\cdots 18}{20\cdots 99}a^{9}+\frac{43\cdots 04}{188773156896909}a^{8}+\frac{42\cdots 82}{62\cdots 97}a^{7}-\frac{868773811844896}{62924385632303}a^{6}+\frac{61\cdots 08}{62\cdots 97}a^{5}-\frac{39\cdots 85}{122147336815647}a^{4}-\frac{93\cdots 68}{692168241955333}a^{3}+\frac{53\cdots 91}{20\cdots 99}a^{2}-\frac{217096978138779}{62924385632303}a+\frac{11\cdots 17}{692168241955333}$, $\frac{87395385399743}{479193398276769}a^{17}-\frac{443591422008884}{479193398276769}a^{16}+\frac{796852420347928}{479193398276769}a^{15}-\frac{217670979466366}{479193398276769}a^{14}-\frac{134955349342399}{159731132758923}a^{13}+\frac{242144130517913}{9395948985819}a^{12}-\frac{16\cdots 82}{43563036206979}a^{11}+\frac{59\cdots 18}{479193398276769}a^{10}-\frac{10\cdots 51}{159731132758923}a^{9}+\frac{28\cdots 24}{159731132758923}a^{8}+\frac{23\cdots 54}{43563036206979}a^{7}-\frac{76875269272495}{479193398276769}a^{6}+\frac{16\cdots 73}{479193398276769}a^{5}-\frac{66\cdots 85}{43563036206979}a^{4}-\frac{66\cdots 19}{53243710919641}a^{3}+\frac{386276050515896}{3895881286803}a^{2}+\frac{314928330806183}{53243710919641}a-\frac{282985004427882}{53243710919641}$ Copy content Toggle raw display (assuming GRH)
Copy content comment:Fundamental units
 
Copy content sage:UK.fundamental_units()
 
Copy content gp:K.fu
 
Copy content magma:[K|fUK(g): g in Generators(UK)];
 
Copy content oscar:[K(fUK(a)) for a in gens(UK)]
 
Regulator:  \( 48971143.43571596 \) (assuming GRH)
Copy content comment:Regulator
 
Copy content sage:K.regulator()
 
Copy content gp:K.reg
 
Copy content magma:Regulator(K);
 
Copy content oscar:regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{8}\cdot 48971143.43571596 \cdot 6}{2\cdot\sqrt{1115906277282951168000000000000}}\cr\approx \mathstrut & 1.35128382160755 \end{aligned}\] (assuming GRH)

Copy content comment:Analytic class number formula
 
Copy content sage:# self-contained SageMath code snippet to compute the analytic class number formula x = polygen(QQ); K.<a> = NumberField(x^18 - 6*x^17 + 14*x^16 - 12*x^15 + 144*x^13 - 335*x^12 + 900*x^11 - 1050*x^10 + 1470*x^9 - 760*x^8 - 36*x^7 + 127*x^6 - 1038*x^5 + 132*x^4 + 990*x^3 - 504*x^2 + 108*x - 27) DK = K.disc(); r1,r2 = K.signature(); RK = K.regulator(); RR = RK.parent() hK = K.class_number(); wK = K.unit_group().torsion_generator().order(); 2^r1 * (2*RR(pi))^r2 * RK * hK / (wK * RR(sqrt(abs(DK))))
 
Copy content gp:\\ self-contained Pari/GP code snippet to compute the analytic class number formula K = bnfinit(x^18 - 6*x^17 + 14*x^16 - 12*x^15 + 144*x^13 - 335*x^12 + 900*x^11 - 1050*x^10 + 1470*x^9 - 760*x^8 - 36*x^7 + 127*x^6 - 1038*x^5 + 132*x^4 + 990*x^3 - 504*x^2 + 108*x - 27, 1); [polcoeff (lfunrootres (lfuncreate (K))[1][1][2], -1), 2^K.r1 * (2*Pi)^K.r2 * K.reg * K.no / (K.tu[1] * sqrt (abs (K.disc)))]
 
Copy content magma:/* self-contained Magma code snippet to compute the analytic class number formula */ Qx<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^18 - 6*x^17 + 14*x^16 - 12*x^15 + 144*x^13 - 335*x^12 + 900*x^11 - 1050*x^10 + 1470*x^9 - 760*x^8 - 36*x^7 + 127*x^6 - 1038*x^5 + 132*x^4 + 990*x^3 - 504*x^2 + 108*x - 27); OK := Integers(K); DK := Discriminant(OK); UK, fUK := UnitGroup(OK); clK, fclK := ClassGroup(OK); r1,r2 := Signature(K); RK := Regulator(K); RR := Parent(RK); hK := #clK; wK := #TorsionSubgroup(UK); 2^r1 * (2*Pi(RR))^r2 * RK * hK / (wK * Sqrt(RR!Abs(DK)));
 
Copy content oscar:# self-contained Oscar code snippet to compute the analytic class number formula Qx, x = PolynomialRing(QQ); K, a = NumberField(x^18 - 6*x^17 + 14*x^16 - 12*x^15 + 144*x^13 - 335*x^12 + 900*x^11 - 1050*x^10 + 1470*x^9 - 760*x^8 - 36*x^7 + 127*x^6 - 1038*x^5 + 132*x^4 + 990*x^3 - 504*x^2 + 108*x - 27); OK = ring_of_integers(K); DK = discriminant(OK); UK, fUK = unit_group(OK); clK, fclK = class_group(OK); r1,r2 = signature(K); RK = regulator(K); RR = parent(RK); hK = order(clK); wK = torsion_units_order(K); 2^r1 * (2*pi)^r2 * RK * hK / (wK * sqrt(RR(abs(DK))))
 

Galois group

$C_6:S_3$ (as 18T12):

Copy content comment:Galois group
 
Copy content sage:K.galois_group(type='pari')
 
Copy content gp:polgalois(K.pol)
 
Copy content magma:G = GaloisGroup(K);
 
Copy content oscar:G, Gtx = galois_group(K); G, transitive_group_identification(G)
 
A solvable group of order 36
The 12 conjugacy class representatives for $C_6:S_3$
Character table for $C_6:S_3$

Intermediate fields

\(\Q(\sqrt{3}) \), 3.1.3675.1, 3.1.588.1, 3.1.14700.1, 3.1.300.1, 6.2.16595712.3, 6.2.2593080000.5, 6.2.10372320000.2, 6.2.4320000.2, 9.1.9529569000000.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Copy content comment:Intermediate fields
 
Copy content sage:K.subfields()[1:-1]
 
Copy content gp:L = nfsubfields(K); L[2..length(b)]
 
Copy content magma:L := Subfields(K); L[2..#L];
 
Copy content oscar:subfields(K)[2:end-1]
 

Sibling fields

Galois closure: data not computed
Degree 18 sibling: 18.0.371968759094317056000000000000.1
Minimal sibling: 18.0.371968759094317056000000000000.1

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R R R R ${\href{/padicField/11.2.0.1}{2} }^{8}{,}\,{\href{/padicField/11.1.0.1}{1} }^{2}$ ${\href{/padicField/13.3.0.1}{3} }^{6}$ ${\href{/padicField/17.2.0.1}{2} }^{9}$ ${\href{/padicField/19.6.0.1}{6} }^{3}$ ${\href{/padicField/23.2.0.1}{2} }^{8}{,}\,{\href{/padicField/23.1.0.1}{1} }^{2}$ ${\href{/padicField/29.2.0.1}{2} }^{9}$ ${\href{/padicField/31.6.0.1}{6} }^{3}$ ${\href{/padicField/37.3.0.1}{3} }^{6}$ ${\href{/padicField/41.2.0.1}{2} }^{9}$ ${\href{/padicField/43.6.0.1}{6} }^{3}$ ${\href{/padicField/47.2.0.1}{2} }^{8}{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ ${\href{/padicField/53.2.0.1}{2} }^{9}$ ${\href{/padicField/59.2.0.1}{2} }^{8}{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

Copy content comment:Frobenius cycle types
 
Copy content sage:# to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Sage: p = 7; [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
Copy content gp:\\ to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Pari: p = 7; pfac = idealprimedec(K, p); vector(length(pfac), j, [pfac[j][3], pfac[j][4]])
 
Copy content magma:// to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Magma: p := 7; [<pr[2], Valuation(Norm(pr[1]), p)> : pr in Factorization(p*Integers(K))];
 
Copy content oscar:# to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Oscar: p = 7; pfac = factor(ideal(ring_of_integers(K), p)); [(e, valuation(norm(pr),p)) for (pr,e) in pfac]
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
\(2\) Copy content Toggle raw display 2.1.6.8a1.2$x^{6} + 2 x^{3} + 6$$6$$1$$8$$D_{6}$$$[2]_{3}^{2}$$
2.2.6.16a1.5$x^{12} + 6 x^{11} + 21 x^{10} + 50 x^{9} + 90 x^{8} + 126 x^{7} + 143 x^{6} + 132 x^{5} + 102 x^{4} + 64 x^{3} + 33 x^{2} + 12 x + 5$$6$$2$$16$$D_6$$$[2]_{3}^{2}$$
\(3\) Copy content Toggle raw display 3.1.2.1a1.2$x^{2} + 6$$2$$1$$1$$C_2$$$[\ ]_{2}$$
3.2.2.2a1.2$x^{4} + 4 x^{3} + 8 x^{2} + 8 x + 7$$2$$2$$2$$C_2^2$$$[\ ]_{2}^{2}$$
3.2.2.2a1.2$x^{4} + 4 x^{3} + 8 x^{2} + 8 x + 7$$2$$2$$2$$C_2^2$$$[\ ]_{2}^{2}$$
3.2.2.2a1.2$x^{4} + 4 x^{3} + 8 x^{2} + 8 x + 7$$2$$2$$2$$C_2^2$$$[\ ]_{2}^{2}$$
3.2.2.2a1.2$x^{4} + 4 x^{3} + 8 x^{2} + 8 x + 7$$2$$2$$2$$C_2^2$$$[\ ]_{2}^{2}$$
\(5\) Copy content Toggle raw display 5.2.3.4a1.2$x^{6} + 12 x^{5} + 54 x^{4} + 112 x^{3} + 108 x^{2} + 48 x + 13$$3$$2$$4$$S_3$$$[\ ]_{3}^{2}$$
5.2.3.4a1.2$x^{6} + 12 x^{5} + 54 x^{4} + 112 x^{3} + 108 x^{2} + 48 x + 13$$3$$2$$4$$S_3$$$[\ ]_{3}^{2}$$
5.2.3.4a1.2$x^{6} + 12 x^{5} + 54 x^{4} + 112 x^{3} + 108 x^{2} + 48 x + 13$$3$$2$$4$$S_3$$$[\ ]_{3}^{2}$$
\(7\) Copy content Toggle raw display 7.6.3.12a1.3$x^{18} + 3 x^{16} + 15 x^{15} + 15 x^{14} + 48 x^{13} + 109 x^{12} + 171 x^{11} + 333 x^{10} + 497 x^{9} + 717 x^{8} + 1032 x^{7} + 1216 x^{6} + 1296 x^{5} + 1143 x^{4} + 783 x^{3} + 432 x^{2} + 162 x + 34$$3$$6$$12$$C_6 \times C_3$$$[\ ]_{3}^{6}$$

Spectrum of ring of integers

(0)(0)(2)(3)(5)(7)(11)(13)(17)(19)(23)(29)(31)(37)(41)(43)(47)(53)(59)